Asset Allocation Efficient Frontier

CFA Level III
#1

In the asset allocation section they discuss the mean-variance method and they make a few claims which I am having trouble rectifying in my head.

First they say that for an unconstrained MV, any two portfolios on the MV frontier define the entire curve - where any other MV portfolio is a linear combination of the two known portfolios … Is this true or is this an appoximation?

The reason I am dubious is that if two portfolios on the frontier have a 0 allocation to a given asset, then all portfolios MUST have a 0 allocation to that asset.

Later on they impose a non-negative weight constraint (no shorts) and then define corner portfolios in that context. Are corner portfolios only applicable to the non-negative weight constrained MV?

Thanks in advance to all for the help.

#2

First they say that for an unconstrained MV, any two portfolios on the MV frontier define the entire curve - where any other MV portfolio is a linear combination of the two known portfolios … Is this true or is this an appoximation?

  • I think what they are trying to say here is that any portfolio on the MV frontier can be expressed as a linear combination of two other known portfolios appearing anywhere on the curve. (lets call this #1)

Later on they impose a non-negative weight constraint (no shorts) and then define corner portfolios in that context. Are corner portfolios only applicable to the non-negative weight constrained MV?

  • Corner portfolios are defined as that portfolios which have only positive weights and still lie on the MV efficient frontier. So using the hypothesis given in #1, we can say that we can represent every portfolio between these two corner portfolios as a linear combination of them. This must answer your second question right?

Hope this helps.

#3

Thanks Sooraj.

Yes, I understand this.

What I was asking was whether this ONLY applied to the unconstrained MV frontier, and I think it does. I’ll circle back to why this can’t apply to the sign constrained frontier.

This is not correct or, at least not stated clearly enough. Corner portfolios are NOT the portfolios on the unconstrained frontier that happen to only have positive weights and still lie on the frontier. Rather, a NEW frontier is derived from those portfolios that are MV optimized with the constraint of only having positive (or zero) allocation. The corner portfolios are the portfolios that, under the constraint of positive allocations, see a change in a particular investment change from 0 to + or + to 0.

According to the answer key on reading 18 question 9A, “Corner portfolios arise from a mean–variance optimization in which asset-class weights are constrained to be nonnegative.”

So the corner portfolio concept does NOT apply to the unconstrained MV frontier.

Anyway, I will look further into the Black two-fund theorem and see if I can understand WHY it is the case that any MV portfolio is a linear combination of any two MV portfolios.

#4

ALSO… see this

The linear approximation just illustrated provides a quick approximation (and upper limit) for the standard deviation; we also can apply this approximation in other cases in which we calculate efficient portfolios using the corner portfolio theorem. (Institute 61) Institute, CFA. 2015 CFA Level III Volume 3 Economic Analysis and Asset Allocation. Wiley Global Finance, 2014-07-14. VitalBook file.

#5

This last part is related to this

if you had two portfolio components (w1, s1) and (w2,s2) with r as the correlation coeff on the returns

actual std deviation = sqrt(w1^2*s1^2 + w2^2*s2^2 + 2*w1*w2*r*s1*s2)

however an approximation is made in the analysis

and they make

std deviation = w1*s1 + w2 * s2 -> which means corr coeff = 1

so the std deviation is an approximation - and it is an upper limit (as r = 1 by the assumption)